Diagrams.Solve.Tridiagonal:solveTriDiagonal from diagrams-solve-0.1, B

Percentage Accurate: 95.7% → 95.7%

Time: 7.9s

Alternatives: 6

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \frac{x}{y - z \cdot t} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (/ x (- y (* z t))))

C

double code(double x, double y, double z, double t) {
	return x / (y - (z * t));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x / (y - (z * t))
end function

Java

public static double code(double x, double y, double z, double t) {
	return x / (y - (z * t));
}

Python

def code(x, y, z, t):
	return x / (y - (z * t))

Julia

function code(x, y, z, t)
	return Float64(x / Float64(y - Float64(z * t)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = x / (y - (z * t));
end

Wolfram

code[x_, y_, z_, t_] := N[(x / N[(y - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 95.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x}{y - z \cdot t} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (/ x (- y (* z t))))

C

double code(double x, double y, double z, double t) {
	return x / (y - (z * t));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x / (y - (z * t))
end function

Java

public static double code(double x, double y, double z, double t) {
	return x / (y - (z * t));
}

Python

def code(x, y, z, t):
	return x / (y - (z * t))

Julia

function code(x, y, z, t)
	return Float64(x / Float64(y - Float64(z * t)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = x / (y - (z * t));
end

Wolfram

code[x_, y_, z_, t_] := N[(x / N[(y - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 95.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x}{y - z \cdot t} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (/ x (- y (* z t))))

C

double code(double x, double y, double z, double t) {
	return x / (y - (z * t));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x / (y - (z * t))
end function

Java

public static double code(double x, double y, double z, double t) {
	return x / (y - (z * t));
}

Python

def code(x, y, z, t):
	return x / (y - (z * t))

Julia

function code(x, y, z, t)
	return Float64(x / Float64(y - Float64(z * t)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = x / (y - (z * t));
end

Wolfram

code[x_, y_, z_, t_] := N[(x / N[(y - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.0%

   \[\frac{x}{y - z \cdot t} \]
2. Add Preprocessing

3. Final simplification 98.0%

   \[\leadsto \frac{x}{y - z \cdot t} \]
4. Add Preprocessing

Alternative 2: 66.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{t \cdot \left(-z\right)}\\ \mathbf{if}\;z \leq -2.2 \cdot 10^{+158}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -1.45 \cdot 10^{+153}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;z \leq -2.1 \cdot 10^{+136}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -1.1 \cdot 10^{+102} \lor \neg \left(z \leq -7.6 \cdot 10^{+29}\right) \land z \leq 8 \cdot 10^{-156}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t}}{-z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ x (* t (- z)))))
   (if (<= z -2.2e+158)
     t_1
     (if (<= z -1.45e+153)
       (/ x y)
       (if (<= z -2.1e+136)
         t_1
         (if (or (<= z -1.1e+102) (and (not (<= z -7.6e+29)) (<= z 8e-156)))
           (/ x y)
           (/ (/ x t) (- z))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x / (t * -z);
	double tmp;
	if (z <= -2.2e+158) {
		tmp = t_1;
	} else if (z <= -1.45e+153) {
		tmp = x / y;
	} else if (z <= -2.1e+136) {
		tmp = t_1;
	} else if ((z <= -1.1e+102) || (!(z <= -7.6e+29) && (z <= 8e-156))) {
		tmp = x / y;
	} else {
		tmp = (x / t) / -z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x / (t * -z)
    if (z <= (-2.2d+158)) then
        tmp = t_1
    else if (z <= (-1.45d+153)) then
        tmp = x / y
    else if (z <= (-2.1d+136)) then
        tmp = t_1
    else if ((z <= (-1.1d+102)) .or. (.not. (z <= (-7.6d+29))) .and. (z <= 8d-156)) then
        tmp = x / y
    else
        tmp = (x / t) / -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x / (t * -z);
	double tmp;
	if (z <= -2.2e+158) {
		tmp = t_1;
	} else if (z <= -1.45e+153) {
		tmp = x / y;
	} else if (z <= -2.1e+136) {
		tmp = t_1;
	} else if ((z <= -1.1e+102) || (!(z <= -7.6e+29) && (z <= 8e-156))) {
		tmp = x / y;
	} else {
		tmp = (x / t) / -z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x / (t * -z)
	tmp = 0
	if z <= -2.2e+158:
		tmp = t_1
	elif z <= -1.45e+153:
		tmp = x / y
	elif z <= -2.1e+136:
		tmp = t_1
	elif (z <= -1.1e+102) or (not (z <= -7.6e+29) and (z <= 8e-156)):
		tmp = x / y
	else:
		tmp = (x / t) / -z
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x / Float64(t * Float64(-z)))
	tmp = 0.0
	if (z <= -2.2e+158)
		tmp = t_1;
	elseif (z <= -1.45e+153)
		tmp = Float64(x / y);
	elseif (z <= -2.1e+136)
		tmp = t_1;
	elseif ((z <= -1.1e+102) || (!(z <= -7.6e+29) && (z <= 8e-156)))
		tmp = Float64(x / y);
	else
		tmp = Float64(Float64(x / t) / Float64(-z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x / (t * -z);
	tmp = 0.0;
	if (z <= -2.2e+158)
		tmp = t_1;
	elseif (z <= -1.45e+153)
		tmp = x / y;
	elseif (z <= -2.1e+136)
		tmp = t_1;
	elseif ((z <= -1.1e+102) || (~((z <= -7.6e+29)) && (z <= 8e-156)))
		tmp = x / y;
	else
		tmp = (x / t) / -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x / N[(t * (-z)), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -2.2e+158], t$95$1, If[LessEqual[z, -1.45e+153], N[(x / y), $MachinePrecision], If[LessEqual[z, -2.1e+136], t$95$1, If[Or[LessEqual[z, -1.1e+102], And[N[Not[LessEqual[z, -7.6e+29]], $MachinePrecision], LessEqual[z, 8e-156]]], N[(x / y), $MachinePrecision], N[(N[(x / t), $MachinePrecision] / (-z)), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.2000000000000001e158 or -1.45000000000000001e153 < z < -2.0999999999999999e136

   1. Initial program 96.7%

      \[\frac{x}{y - z \cdot t} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 79.1%

      \[\leadsto \color{blue}{-1 \cdot \frac{x}{t \cdot z}} \]
   4. Step-by-step derivation

      1. associate-*r/ 79.1%

         \[\leadsto \color{blue}{\frac{-1 \cdot x}{t \cdot z}} \]

      2. neg-mul-1 79.1%

         \[\leadsto \frac{\color{blue}{-x}}{t \cdot z} \]

   5. Simplified 79.1%

      \[\leadsto \color{blue}{\frac{-x}{t \cdot z}} \]

   if -2.2000000000000001e158 < z < -1.45000000000000001e153 or -2.0999999999999999e136 < z < -1.10000000000000004e102 or -7.59999999999999942e29 < z < 8.00000000000000032e-156

   1. Initial program 99.9%

      \[\frac{x}{y - z \cdot t} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 77.5%

      \[\leadsto \color{blue}{\frac{x}{y}} \]

   if -1.10000000000000004e102 < z < -7.59999999999999942e29 or 8.00000000000000032e-156 < z

   1. Initial program 96.4%

      \[\frac{x}{y - z \cdot t} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 60.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{x}{t \cdot z}} \]
   4. Step-by-step derivation

      1. associate-*r/ 60.8%

         \[\leadsto \color{blue}{\frac{-1 \cdot x}{t \cdot z}} \]

      2. neg-mul-1 60.8%

         \[\leadsto \frac{\color{blue}{-x}}{t \cdot z} \]

   5. Simplified 60.8%

      \[\leadsto \color{blue}{\frac{-x}{t \cdot z}} \]
   6. Step-by-step derivation

      1. distribute-frac-neg 60.8%

         \[\leadsto \color{blue}{-\frac{x}{t \cdot z}} \]

      2. *-commutative 60.8%

         \[\leadsto -\frac{x}{\color{blue}{z \cdot t}} \]

      3. associate-/l/ 61.2%

         \[\leadsto -\color{blue}{\frac{\frac{x}{t}}{z}} \]

   7. Applied egg-rr 61.2%

      \[\leadsto \color{blue}{-\frac{\frac{x}{t}}{z}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.2 \cdot 10^{+158}:\\ \;\;\;\;\frac{x}{t \cdot \left(-z\right)}\\ \mathbf{elif}\;z \leq -1.45 \cdot 10^{+153}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;z \leq -2.1 \cdot 10^{+136}:\\ \;\;\;\;\frac{x}{t \cdot \left(-z\right)}\\ \mathbf{elif}\;z \leq -1.1 \cdot 10^{+102} \lor \neg \left(z \leq -7.6 \cdot 10^{+29}\right) \land z \leq 8 \cdot 10^{-156}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t}}{-z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 66.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -7.8 \cdot 10^{+158}:\\ \;\;\;\;x \cdot \frac{\frac{-1}{z}}{t}\\ \mathbf{elif}\;z \leq -1.45 \cdot 10^{+153}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;z \leq -1.95 \cdot 10^{+136}:\\ \;\;\;\;\frac{\frac{x}{-z}}{t}\\ \mathbf{elif}\;z \leq -2.15 \cdot 10^{+96} \lor \neg \left(z \leq -1.05 \cdot 10^{+31}\right) \land z \leq 8 \cdot 10^{-156}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t}}{-z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -7.8e+158)
   (* x (/ (/ -1.0 z) t))
   (if (<= z -1.45e+153)
     (/ x y)
     (if (<= z -1.95e+136)
       (/ (/ x (- z)) t)
       (if (or (<= z -2.15e+96) (and (not (<= z -1.05e+31)) (<= z 8e-156)))
         (/ x y)
         (/ (/ x t) (- z)))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -7.8e+158) {
		tmp = x * ((-1.0 / z) / t);
	} else if (z <= -1.45e+153) {
		tmp = x / y;
	} else if (z <= -1.95e+136) {
		tmp = (x / -z) / t;
	} else if ((z <= -2.15e+96) || (!(z <= -1.05e+31) && (z <= 8e-156))) {
		tmp = x / y;
	} else {
		tmp = (x / t) / -z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= (-7.8d+158)) then
        tmp = x * (((-1.0d0) / z) / t)
    else if (z <= (-1.45d+153)) then
        tmp = x / y
    else if (z <= (-1.95d+136)) then
        tmp = (x / -z) / t
    else if ((z <= (-2.15d+96)) .or. (.not. (z <= (-1.05d+31))) .and. (z <= 8d-156)) then
        tmp = x / y
    else
        tmp = (x / t) / -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -7.8e+158) {
		tmp = x * ((-1.0 / z) / t);
	} else if (z <= -1.45e+153) {
		tmp = x / y;
	} else if (z <= -1.95e+136) {
		tmp = (x / -z) / t;
	} else if ((z <= -2.15e+96) || (!(z <= -1.05e+31) && (z <= 8e-156))) {
		tmp = x / y;
	} else {
		tmp = (x / t) / -z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -7.8e+158:
		tmp = x * ((-1.0 / z) / t)
	elif z <= -1.45e+153:
		tmp = x / y
	elif z <= -1.95e+136:
		tmp = (x / -z) / t
	elif (z <= -2.15e+96) or (not (z <= -1.05e+31) and (z <= 8e-156)):
		tmp = x / y
	else:
		tmp = (x / t) / -z
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -7.8e+158)
		tmp = Float64(x * Float64(Float64(-1.0 / z) / t));
	elseif (z <= -1.45e+153)
		tmp = Float64(x / y);
	elseif (z <= -1.95e+136)
		tmp = Float64(Float64(x / Float64(-z)) / t);
	elseif ((z <= -2.15e+96) || (!(z <= -1.05e+31) && (z <= 8e-156)))
		tmp = Float64(x / y);
	else
		tmp = Float64(Float64(x / t) / Float64(-z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -7.8e+158)
		tmp = x * ((-1.0 / z) / t);
	elseif (z <= -1.45e+153)
		tmp = x / y;
	elseif (z <= -1.95e+136)
		tmp = (x / -z) / t;
	elseif ((z <= -2.15e+96) || (~((z <= -1.05e+31)) && (z <= 8e-156)))
		tmp = x / y;
	else
		tmp = (x / t) / -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -7.8e+158], N[(x * N[(N[(-1.0 / z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, -1.45e+153], N[(x / y), $MachinePrecision], If[LessEqual[z, -1.95e+136], N[(N[(x / (-z)), $MachinePrecision] / t), $MachinePrecision], If[Or[LessEqual[z, -2.15e+96], And[N[Not[LessEqual[z, -1.05e+31]], $MachinePrecision], LessEqual[z, 8e-156]]], N[(x / y), $MachinePrecision], N[(N[(x / t), $MachinePrecision] / (-z)), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -7.8e158

   1. Initial program 96.5%

      \[\frac{x}{y - z \cdot t} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf 89.9%

      \[\leadsto \frac{x}{\color{blue}{t \cdot \left(\frac{y}{t} - z\right)}} \]
   4. Step-by-step derivation

      1. div-inv 89.8%

         \[\leadsto \color{blue}{x \cdot \frac{1}{t \cdot \left(\frac{y}{t} - z\right)}} \]

   5. Applied egg-rr 89.8%

      \[\leadsto \color{blue}{x \cdot \frac{1}{t \cdot \left(\frac{y}{t} - z\right)}} \]

   6. Taylor expanded in t around inf 77.6%

      \[\leadsto x \cdot \color{blue}{\frac{-1}{t \cdot z}} \]
   7. Step-by-step derivation

      1. metadata-eval 77.6%

         \[\leadsto x \cdot \frac{\color{blue}{-1}}{t \cdot z} \]

      2. *-commutative 77.6%

         \[\leadsto x \cdot \frac{-1}{\color{blue}{z \cdot t}} \]

      3. distribute-neg-frac 77.6%

         \[\leadsto x \cdot \color{blue}{\left(-\frac{1}{z \cdot t}\right)} \]

      4. associate-/r* 80.7%

         \[\leadsto x \cdot \left(-\color{blue}{\frac{\frac{1}{z}}{t}}\right) \]

      5. distribute-neg-frac 80.7%

         \[\leadsto x \cdot \color{blue}{\frac{-\frac{1}{z}}{t}} \]

      6. distribute-neg-frac 80.7%

         \[\leadsto x \cdot \frac{\color{blue}{\frac{-1}{z}}}{t} \]

      7. metadata-eval 80.7%

         \[\leadsto x \cdot \frac{\frac{\color{blue}{-1}}{z}}{t} \]

   8. Simplified 80.7%

      \[\leadsto x \cdot \color{blue}{\frac{\frac{-1}{z}}{t}} \]

   if -7.8e158 < z < -1.45000000000000001e153 or -1.9500000000000001e136 < z < -2.15000000000000001e96 or -1.04999999999999989e31 < z < 8.00000000000000032e-156

   1. Initial program 99.9%

      \[\frac{x}{y - z \cdot t} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 77.5%

      \[\leadsto \color{blue}{\frac{x}{y}} \]

   if -1.45000000000000001e153 < z < -1.9500000000000001e136

   1. Initial program 99.2%

      \[\frac{x}{y - z \cdot t} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf 99.2%

      \[\leadsto \frac{x}{\color{blue}{t \cdot \left(\frac{y}{t} - z\right)}} \]
   4. Step-by-step derivation

      1. div-inv 99.2%

         \[\leadsto \color{blue}{x \cdot \frac{1}{t \cdot \left(\frac{y}{t} - z\right)}} \]

   5. Applied egg-rr 99.2%

      \[\leadsto \color{blue}{x \cdot \frac{1}{t \cdot \left(\frac{y}{t} - z\right)}} \]

   6. Taylor expanded in t around inf 99.2%

      \[\leadsto x \cdot \color{blue}{\frac{-1}{t \cdot z}} \]
   7. Step-by-step derivation

      1. metadata-eval 99.2%

         \[\leadsto x \cdot \frac{\color{blue}{-1}}{t \cdot z} \]

      2. *-commutative 99.2%

         \[\leadsto x \cdot \frac{-1}{\color{blue}{z \cdot t}} \]

      3. distribute-neg-frac 99.2%

         \[\leadsto x \cdot \color{blue}{\left(-\frac{1}{z \cdot t}\right)} \]

      4. associate-/r* 99.2%

         \[\leadsto x \cdot \left(-\color{blue}{\frac{\frac{1}{z}}{t}}\right) \]

      5. distribute-neg-frac 99.2%

         \[\leadsto x \cdot \color{blue}{\frac{-\frac{1}{z}}{t}} \]

      6. distribute-neg-frac 99.2%

         \[\leadsto x \cdot \frac{\color{blue}{\frac{-1}{z}}}{t} \]

      7. metadata-eval 99.2%

         \[\leadsto x \cdot \frac{\frac{\color{blue}{-1}}{z}}{t} \]

   8. Simplified 99.2%

      \[\leadsto x \cdot \color{blue}{\frac{\frac{-1}{z}}{t}} \]
   9. Step-by-step derivation

      1. associate-*r/ 99.2%

         \[\leadsto \color{blue}{\frac{x \cdot \frac{-1}{z}}{t}} \]

      2. frac-2neg 99.2%

         \[\leadsto \frac{x \cdot \color{blue}{\frac{--1}{-z}}}{t} \]

      3. metadata-eval 99.2%

         \[\leadsto \frac{x \cdot \frac{\color{blue}{1}}{-z}}{t} \]

      4. un-div-inv 100.0%

         \[\leadsto \frac{\color{blue}{\frac{x}{-z}}}{t} \]

   10. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{\frac{\frac{x}{-z}}{t}} \]

   if -2.15000000000000001e96 < z < -1.04999999999999989e31 or 8.00000000000000032e-156 < z

   1. Initial program 96.4%

      \[\frac{x}{y - z \cdot t} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 60.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{x}{t \cdot z}} \]
   4. Step-by-step derivation

      1. associate-*r/ 60.8%

         \[\leadsto \color{blue}{\frac{-1 \cdot x}{t \cdot z}} \]

      2. neg-mul-1 60.8%

         \[\leadsto \frac{\color{blue}{-x}}{t \cdot z} \]

   5. Simplified 60.8%

      \[\leadsto \color{blue}{\frac{-x}{t \cdot z}} \]
   6. Step-by-step derivation

      1. distribute-frac-neg 60.8%

         \[\leadsto \color{blue}{-\frac{x}{t \cdot z}} \]

      2. *-commutative 60.8%

         \[\leadsto -\frac{x}{\color{blue}{z \cdot t}} \]

      3. associate-/l/ 61.2%

         \[\leadsto -\color{blue}{\frac{\frac{x}{t}}{z}} \]

   7. Applied egg-rr 61.2%

      \[\leadsto \color{blue}{-\frac{\frac{x}{t}}{z}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.8 \cdot 10^{+158}:\\ \;\;\;\;x \cdot \frac{\frac{-1}{z}}{t}\\ \mathbf{elif}\;z \leq -1.45 \cdot 10^{+153}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;z \leq -1.95 \cdot 10^{+136}:\\ \;\;\;\;\frac{\frac{x}{-z}}{t}\\ \mathbf{elif}\;z \leq -2.15 \cdot 10^{+96} \lor \neg \left(z \leq -1.05 \cdot 10^{+31}\right) \land z \leq 8 \cdot 10^{-156}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t}}{-z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 67.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.6 \cdot 10^{+136}:\\ \;\;\;\;\frac{\frac{x}{-z}}{t}\\ \mathbf{elif}\;z \leq -2.5 \cdot 10^{+102} \lor \neg \left(z \leq -2.05 \cdot 10^{+33}\right) \land z \leq 8 \cdot 10^{-156}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t}}{-z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -2.6e+136)
   (/ (/ x (- z)) t)
   (if (or (<= z -2.5e+102) (and (not (<= z -2.05e+33)) (<= z 8e-156)))
     (/ x y)
     (/ (/ x t) (- z)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -2.6e+136) {
		tmp = (x / -z) / t;
	} else if ((z <= -2.5e+102) || (!(z <= -2.05e+33) && (z <= 8e-156))) {
		tmp = x / y;
	} else {
		tmp = (x / t) / -z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= (-2.6d+136)) then
        tmp = (x / -z) / t
    else if ((z <= (-2.5d+102)) .or. (.not. (z <= (-2.05d+33))) .and. (z <= 8d-156)) then
        tmp = x / y
    else
        tmp = (x / t) / -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -2.6e+136) {
		tmp = (x / -z) / t;
	} else if ((z <= -2.5e+102) || (!(z <= -2.05e+33) && (z <= 8e-156))) {
		tmp = x / y;
	} else {
		tmp = (x / t) / -z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -2.6e+136:
		tmp = (x / -z) / t
	elif (z <= -2.5e+102) or (not (z <= -2.05e+33) and (z <= 8e-156)):
		tmp = x / y
	else:
		tmp = (x / t) / -z
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -2.6e+136)
		tmp = Float64(Float64(x / Float64(-z)) / t);
	elseif ((z <= -2.5e+102) || (!(z <= -2.05e+33) && (z <= 8e-156)))
		tmp = Float64(x / y);
	else
		tmp = Float64(Float64(x / t) / Float64(-z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -2.6e+136)
		tmp = (x / -z) / t;
	elseif ((z <= -2.5e+102) || (~((z <= -2.05e+33)) && (z <= 8e-156)))
		tmp = x / y;
	else
		tmp = (x / t) / -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -2.6e+136], N[(N[(x / (-z)), $MachinePrecision] / t), $MachinePrecision], If[Or[LessEqual[z, -2.5e+102], And[N[Not[LessEqual[z, -2.05e+33]], $MachinePrecision], LessEqual[z, 8e-156]]], N[(x / y), $MachinePrecision], N[(N[(x / t), $MachinePrecision] / (-z)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.6000000000000001e136

   1. Initial program 96.8%

      \[\frac{x}{y - z \cdot t} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf 87.9%

      \[\leadsto \frac{x}{\color{blue}{t \cdot \left(\frac{y}{t} - z\right)}} \]
   4. Step-by-step derivation

      1. div-inv 87.8%

         \[\leadsto \color{blue}{x \cdot \frac{1}{t \cdot \left(\frac{y}{t} - z\right)}} \]

   5. Applied egg-rr 87.8%

      \[\leadsto \color{blue}{x \cdot \frac{1}{t \cdot \left(\frac{y}{t} - z\right)}} \]

   6. Taylor expanded in t around inf 76.6%

      \[\leadsto x \cdot \color{blue}{\frac{-1}{t \cdot z}} \]
   7. Step-by-step derivation

      1. metadata-eval 76.6%

         \[\leadsto x \cdot \frac{\color{blue}{-1}}{t \cdot z} \]

      2. *-commutative 76.6%

         \[\leadsto x \cdot \frac{-1}{\color{blue}{z \cdot t}} \]

      3. distribute-neg-frac 76.6%

         \[\leadsto x \cdot \color{blue}{\left(-\frac{1}{z \cdot t}\right)} \]

      4. associate-/r* 79.4%

         \[\leadsto x \cdot \left(-\color{blue}{\frac{\frac{1}{z}}{t}}\right) \]

      5. distribute-neg-frac 79.4%

         \[\leadsto x \cdot \color{blue}{\frac{-\frac{1}{z}}{t}} \]

      6. distribute-neg-frac 79.4%

         \[\leadsto x \cdot \frac{\color{blue}{\frac{-1}{z}}}{t} \]

      7. metadata-eval 79.4%

         \[\leadsto x \cdot \frac{\frac{\color{blue}{-1}}{z}}{t} \]

   8. Simplified 79.4%

      \[\leadsto x \cdot \color{blue}{\frac{\frac{-1}{z}}{t}} \]
   9. Step-by-step derivation

      1. associate-*r/ 79.7%

         \[\leadsto \color{blue}{\frac{x \cdot \frac{-1}{z}}{t}} \]

      2. frac-2neg 79.7%

         \[\leadsto \frac{x \cdot \color{blue}{\frac{--1}{-z}}}{t} \]

      3. metadata-eval 79.7%

         \[\leadsto \frac{x \cdot \frac{\color{blue}{1}}{-z}}{t} \]

      4. un-div-inv 79.6%

         \[\leadsto \frac{\color{blue}{\frac{x}{-z}}}{t} \]

   10. Applied egg-rr 79.6%

       \[\leadsto \color{blue}{\frac{\frac{x}{-z}}{t}} \]

   if -2.6000000000000001e136 < z < -2.5e102 or -2.04999999999999997e33 < z < 8.00000000000000032e-156

   1. Initial program 99.9%

      \[\frac{x}{y - z \cdot t} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 77.3%

      \[\leadsto \color{blue}{\frac{x}{y}} \]

   if -2.5e102 < z < -2.04999999999999997e33 or 8.00000000000000032e-156 < z

   1. Initial program 96.4%

      \[\frac{x}{y - z \cdot t} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 60.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{x}{t \cdot z}} \]
   4. Step-by-step derivation

      1. associate-*r/ 60.8%

         \[\leadsto \color{blue}{\frac{-1 \cdot x}{t \cdot z}} \]

      2. neg-mul-1 60.8%

         \[\leadsto \frac{\color{blue}{-x}}{t \cdot z} \]

   5. Simplified 60.8%

      \[\leadsto \color{blue}{\frac{-x}{t \cdot z}} \]
   6. Step-by-step derivation

      1. distribute-frac-neg 60.8%

         \[\leadsto \color{blue}{-\frac{x}{t \cdot z}} \]

      2. *-commutative 60.8%

         \[\leadsto -\frac{x}{\color{blue}{z \cdot t}} \]

      3. associate-/l/ 61.2%

         \[\leadsto -\color{blue}{\frac{\frac{x}{t}}{z}} \]

   7. Applied egg-rr 61.2%

      \[\leadsto \color{blue}{-\frac{\frac{x}{t}}{z}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.6 \cdot 10^{+136}:\\ \;\;\;\;\frac{\frac{x}{-z}}{t}\\ \mathbf{elif}\;z \leq -2.5 \cdot 10^{+102} \lor \neg \left(z \leq -2.05 \cdot 10^{+33}\right) \land z \leq 8 \cdot 10^{-156}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t}}{-z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 70.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -155000:\\ \;\;\;\;\frac{1}{\frac{y}{x}}\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{-104}:\\ \;\;\;\;\frac{\frac{x}{t}}{-z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -155000.0)
   (/ 1.0 (/ y x))
   (if (<= y 6.5e-104) (/ (/ x t) (- z)) (/ x y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -155000.0) {
		tmp = 1.0 / (y / x);
	} else if (y <= 6.5e-104) {
		tmp = (x / t) / -z;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= (-155000.0d0)) then
        tmp = 1.0d0 / (y / x)
    else if (y <= 6.5d-104) then
        tmp = (x / t) / -z
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -155000.0) {
		tmp = 1.0 / (y / x);
	} else if (y <= 6.5e-104) {
		tmp = (x / t) / -z;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= -155000.0:
		tmp = 1.0 / (y / x)
	elif y <= 6.5e-104:
		tmp = (x / t) / -z
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -155000.0)
		tmp = Float64(1.0 / Float64(y / x));
	elseif (y <= 6.5e-104)
		tmp = Float64(Float64(x / t) / Float64(-z));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= -155000.0)
		tmp = 1.0 / (y / x);
	elseif (y <= 6.5e-104)
		tmp = (x / t) / -z;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -155000.0], N[(1.0 / N[(y / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6.5e-104], N[(N[(x / t), $MachinePrecision] / (-z)), $MachinePrecision], N[(x / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -155000

   1. Initial program 98.2%

      \[\frac{x}{y - z \cdot t} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf 81.0%

      \[\leadsto \frac{x}{\color{blue}{t \cdot \left(\frac{y}{t} - z\right)}} \]
   4. Step-by-step derivation

      1. div-inv 80.8%

         \[\leadsto \color{blue}{x \cdot \frac{1}{t \cdot \left(\frac{y}{t} - z\right)}} \]

   5. Applied egg-rr 80.8%

      \[\leadsto \color{blue}{x \cdot \frac{1}{t \cdot \left(\frac{y}{t} - z\right)}} \]

   6. Taylor expanded in t around 0 80.2%

      \[\leadsto x \cdot \color{blue}{\frac{1}{y}} \]
   7. Step-by-step derivation

      1. un-div-inv 80.6%

         \[\leadsto \color{blue}{\frac{x}{y}} \]

      2. clear-num 80.8%

         \[\leadsto \color{blue}{\frac{1}{\frac{y}{x}}} \]

   8. Applied egg-rr 80.8%

      \[\leadsto \color{blue}{\frac{1}{\frac{y}{x}}} \]

   if -155000 < y < 6.49999999999999991e-104

   1. Initial program 97.0%

      \[\frac{x}{y - z \cdot t} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 74.4%

      \[\leadsto \color{blue}{-1 \cdot \frac{x}{t \cdot z}} \]
   4. Step-by-step derivation

      1. associate-*r/ 74.4%

         \[\leadsto \color{blue}{\frac{-1 \cdot x}{t \cdot z}} \]

      2. neg-mul-1 74.4%

         \[\leadsto \frac{\color{blue}{-x}}{t \cdot z} \]

   5. Simplified 74.4%

      \[\leadsto \color{blue}{\frac{-x}{t \cdot z}} \]
   6. Step-by-step derivation

      1. distribute-frac-neg 74.4%

         \[\leadsto \color{blue}{-\frac{x}{t \cdot z}} \]

      2. *-commutative 74.4%

         \[\leadsto -\frac{x}{\color{blue}{z \cdot t}} \]

      3. associate-/l/ 72.4%

         \[\leadsto -\color{blue}{\frac{\frac{x}{t}}{z}} \]

   7. Applied egg-rr 72.4%

      \[\leadsto \color{blue}{-\frac{\frac{x}{t}}{z}} \]

   if 6.49999999999999991e-104 < y

   1. Initial program 99.0%

      \[\frac{x}{y - z \cdot t} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 73.0%

      \[\leadsto \color{blue}{\frac{x}{y}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -155000:\\ \;\;\;\;\frac{1}{\frac{y}{x}}\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{-104}:\\ \;\;\;\;\frac{\frac{x}{t}}{-z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 54.6% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \frac{x}{y} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (/ x y))

C

double code(double x, double y, double z, double t) {
	return x / y;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x / y
end function

Java

public static double code(double x, double y, double z, double t) {
	return x / y;
}

Python

def code(x, y, z, t):
	return x / y

Julia

function code(x, y, z, t)
	return Float64(x / y)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = x / y;
end

Wolfram

code[x_, y_, z_, t_] := N[(x / y), $MachinePrecision]

Derivation

1. Initial program 98.0%

   \[\frac{x}{y - z \cdot t} \]
2. Add Preprocessing

3. Taylor expanded in y around inf 56.9%

   \[\leadsto \color{blue}{\frac{x}{y}} \]

4. Final simplification 56.9%

   \[\leadsto \frac{x}{y} \]
5. Add Preprocessing

Developer target: 96.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{1}{\frac{y}{x} - \frac{z}{x} \cdot t}\\ \mathbf{if}\;x < -1.618195973607049 \cdot 10^{+50}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x < 2.1378306434876444 \cdot 10^{+131}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ 1.0 (- (/ y x) (* (/ z x) t)))))
   (if (< x -1.618195973607049e+50)
     t_1
     (if (< x 2.1378306434876444e+131) (/ x (- y (* z t))) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = 1.0 / ((y / x) - ((z / x) * t));
	double tmp;
	if (x < -1.618195973607049e+50) {
		tmp = t_1;
	} else if (x < 2.1378306434876444e+131) {
		tmp = x / (y - (z * t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 1.0d0 / ((y / x) - ((z / x) * t))
    if (x < (-1.618195973607049d+50)) then
        tmp = t_1
    else if (x < 2.1378306434876444d+131) then
        tmp = x / (y - (z * t))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = 1.0 / ((y / x) - ((z / x) * t));
	double tmp;
	if (x < -1.618195973607049e+50) {
		tmp = t_1;
	} else if (x < 2.1378306434876444e+131) {
		tmp = x / (y - (z * t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = 1.0 / ((y / x) - ((z / x) * t))
	tmp = 0
	if x < -1.618195973607049e+50:
		tmp = t_1
	elif x < 2.1378306434876444e+131:
		tmp = x / (y - (z * t))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(1.0 / Float64(Float64(y / x) - Float64(Float64(z / x) * t)))
	tmp = 0.0
	if (x < -1.618195973607049e+50)
		tmp = t_1;
	elseif (x < 2.1378306434876444e+131)
		tmp = Float64(x / Float64(y - Float64(z * t)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = 1.0 / ((y / x) - ((z / x) * t));
	tmp = 0.0;
	if (x < -1.618195973607049e+50)
		tmp = t_1;
	elseif (x < 2.1378306434876444e+131)
		tmp = x / (y - (z * t));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(1.0 / N[(N[(y / x), $MachinePrecision] - N[(N[(z / x), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Less[x, -1.618195973607049e+50], t$95$1, If[Less[x, 2.1378306434876444e+131], N[(x / N[(y - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Reproduce

herbie shell --seed 2024115 
(FPCore (x y z t)
  :name "Diagrams.Solve.Tridiagonal:solveTriDiagonal from diagrams-solve-0.1, B"
  :precision binary64

  :alt
  (if (< x -1.618195973607049e+50) (/ 1.0 (- (/ y x) (* (/ z x) t))) (if (< x 2.1378306434876444e+131) (/ x (- y (* z t))) (/ 1.0 (- (/ y x) (* (/ z x) t)))))

  (/ x (- y (* z t))))